Quantifying Model Risk in Option Pricing and Value-at-Risk Models

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Quantifying Model Risk in Option Pricing and

Value-at-Risk Models

Dumisani Ngwenza

A dissertation submitted to the Faculty of Commerce, University of Cape Town, in partial fulfilment of the requirements for the degree of Master of Philosophy.

September 14, 2019 MPhil in Mathematical Finance,

University of Cape Town.

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The copyright of this thesis vests in the author. No

quotation from it or information derived from it is to be

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Declaration

I declare that this dissertation is my own, unaided work. It is being submitted for the Degree of Master of Philosophy to the University of Cape Town. It has not before been submitted for any degree or examination.

Name Surname

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Abstract

Financial practitioners use models in order to price, hedge and measure risk. These models are reliant on assumptions and are prone to ”model risk”. Increased inno-vation in complex financial products has lead to increased risk exposure and has spurred research into understanding model risk and its underlying factors. This dissertation quantifies model risk inherent in Value-at-Risk (VaR) on a variety of portfolios comprised of European options written on the ALSI futures index across various maturities. The European options under consideration will be modelled using the Black-Scholes, Heston and Variance-Gamma models.

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Acknowledgements

I would like to acknowledge Obeid Mahomed and Associate Professor Peter Ouwe-hand for the guidance, patience and invaluable input. I would also like to Associate Professor David Taylor for helping me during the writing period. Lastly I am grate-ful for the support and patience from the AIFMRM department, classmates, friends and family.

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List of Figures

4.1 Comparison of Black-Scholes and market implied call option prices

across various days-to-maturities.. . . 12

4.2 Market implied volatility smile, Heston volatility smile and BS con-stant volatility across various days-to-maturities. . . 14

4.3 Comparison of Heston and market implied call option prices across various days-to-maturities. . . 15

4.4 Market implied volatility smile, VG volatility smile and BS constant volatility across various days-to-maturities. . . 16

4.5 Comparison of VG and market implied call option prices across var-ious days-to-maturities. . . 17

5.1 Distribution of St10 under Historical Bootstrap, Black-Scholes, Hes-ton and Variance Gamma assumptions after 10 days. . . 20

5.2 ∆C for BS, Heston and VG at time t0 . . . 22

6.1 ATM call P&L distributions for BS, Heston and VG 90 with days-to-maturity. . . 26

6.2 Long-only portfolio P&L distributions for BS, Heston and VG with 90days-to-maturity.. . . 27

6.3 Collar portfolio P&L distributions for BS, Heston and VG models with 90 days-to-maturity.. . . 27

6.4 ATM Nominal and Perturbed MC 95% VaR . . . 30

6.5 Long-only Nominal and Perturbed MC 95% VaR . . . 32

6.6 Collar Nominal and Perturbed MC 95% VaR . . . 33

B.1 ATM portfolio nominal and perturbed P&L Distributions for the BS, Heston and VG models 30 days-to-maturity options.. . . 40

B.2 Long-only portfolio nominal and perturbed P&L Distributions for the BS, Heston and VG models 30 days-to-maturity options. . . 41

B.3 Collar portfolio nominal and perturbed P&L Distributions for the BS, Heston and VG models 30 days-to-maturity options.. . . 42

B.4 ATM portfolio perturbed P&L Distributions for the Black-Scholes (BS), Heston and VG models 180 days-to-maturity options. . . 43

B.5 Long-only portfolio perturbed P&L Distributions for the Black-Scholes (BS), Heston and VG models 180 days-to-maturity options. . . 44

B.6 Collar portfolio perturbed P&L Distributions for the Black-Scholes (BS), Heston and VG models 180 days-to-maturity options. . . 45

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B.7 ATM portfolio perturbed P&L Distributions for the Black-Scholes (BS), Heston and VG models 270 days-to-maturity options. . . 46

B.8 Long-only portfolio perturbed P&L Distributions for the Black-Scholes (BS), Heston and VG models 270 days-to-maturity options. . . 47

B.9 Collar portfolio perturbed P&L Distributions for the Black-Scholes (BS), Heston and VG models 270 days-to-maturity options. . . 48

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List of Tables

4.1 Black-Scholes (BS) calibration results. . . 12

4.2 Heston model calibration results . . . 14

4.3 VG calibrated parameters . . . 16

6.1 10-day VaR of portfolios with τ = 90 days-to-maturity. . . 28

B.1 10-day VaR of portfolios with τ = 30 days-to-maturity. . . 48

B.3 10-day VaR of portfolios with τ = 270 days-to-maturity . . . 49

B.2 10-day VaR of portfolios with τ = 180 days-to-maturity . . . 49

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Chapter 1 Introduction

Financial risk measurement is a fundamental practice that relies on the use of mod-els and market information for pricing, hedging, reserving and regulatory pur-poses. The models are often heavily reliant on a variety of assumptions and are thus themselves a source of what is referred to as ”model risk”. Model risk was identified as a key component of the financial crisis of 2008/9. With an increase in risk exposure from both the number and complexity of financial products and their models, quantifying model risk is imperative.

Derman(1996) investigates some properties of model risk, in particular

discern-ing the factors that introduce risk in any model.Detering and Packham(2016)

eval-uate the intrinsic model risk in Value-at-Risk (VaR) and Expected Shortfall when

distributional losses are generated by model-dependent hedging. Kerkhof et al.

(2010) evaluates model risk by directly integrating the model error into VaR.Schl ¨ogl

(2016) presents a framework in which model error is classified into four categories

of risk factors. This framework is essential when distinguishing factors that affect model risk for pricing, hedging or risk measurement.

This dissertation evaluates the model risk in market risk capital charge for a variety of portfolios of European options, using VaR as the risk metric. VaR is con-cisely defined as the expected loss of a portfolio, with a certain level of confidence over a period of time. We evaluate VaR in a model-dependent manner where the profit and loss (P&L) is determined by static delta hedging. VaR is then interpreted as the expected loss if the portfolio cannot be hedged over a certain period of time. In this context, we choose to quantify model risk using the methodology of

Glasserman and Xu(2014). This methodology continues the work ofHansen and Sargent(2008), which is based on theKullback and Leibler(1951) divergence (also referred to as relative entropy). The Kullback-Leibler (KL) divergence measures the additional information required in order for an alternative model to be preferable to a nominal model. By comparing a nominal model to a (possibly) preferable alternative, KL divergence is the foundation for quantifying model risk.

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Chapter 1. Introduction 2

The VaR models we consider are the Historical VaR, as the nominal model, and the Monte Carlo VaR, as the alternative model. These are determined on three portfolios consisting of SAFEX (South African Futures Exchange) European op-tions written on ALSI (All Share Index) futures contracts. The analysis of VaR on three different portfolios will determine whether Monte Carlo VaR and Historical VaR are commensurate. This is crucial for regulatory purposes. A commensurate Monte Carlo VaR may permit the quantification of model risk on market risk cap-ital charges for exotic portfolios. Historical data on these might not be available, hence invalidating Historical VaR.

The three different portfolios that will be analysed consist of a long position in an at-the-money (ATM) call option, a long-only portfolio with three call options of various moneyness, and a collar portfolio consisting of a short position in an out-the-money (OTM) call option, a long position in an OTM put option and a long position in the underlying ALSI futures contract. These are all analysed over vari-ous times-to-maturity. The option pricing models use to value the SAFEX European

options will beBlack and Scholes(1973), which assumes a constant volatility, the

Heston(1993) Stochastic Volatility model, which aims to reproduce the volatility

smile, and the Variance Gamma model pioneered by Madan et al. (1998), which

assumes market information is known at random times.

The rest of the dissertation is as follows. Chapter2 gives an overview of the

model risk methodology ofGlasserman and Xu(2014). In particular, the role of the

KL divergence, which forms the foundation for quantifying the worst-case model error, and the robust Monte Carlo approach to estimating model error will be

dis-cussed. Chapter3 explores the stochastic models that will be applied in the

val-uation of the SAFEX European options. A discussion of the calibration of these

models follows in Chapter4. Chapter5 gives an overview of the Historical and

Monte Carlo VaR models and the delta hedging methodology used to obtain the

profit and loss distributions. Chapter6presents the results for model error.

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Chapter 2 Model Risk

In order to compare models, a tool is required to measure their disparity. The

work of Glasserman and Xu(2014) on model error relies on the use of the

Kull-back and Leibler(1951) divergence (or relative entropy) to measure this disparity. This chapter begins by defining model risk and discussing a framework developed bySchl ¨ogl(2016) on the classification of model risk. This is followed by the quan-tification of model risk, beginning with the definition of the Kullback-Leibler di-vergence and a brief overview of its properties. This is followed by a discussion of

model risk and the robust Monte Carlo approach based on the work ofGlasserman

and Xu(2014).

2.1 Model Risk Classification

The use of financial models are heavily reliant on assumptions and are a source of risk. This risk is termed ”model risk” and has led research on its quantification.

Schl ¨ogl (2016) presents a general framework that classifies model risk into four categories:

1. Type 0 describes the risk of the model associated with parameters. This in-cludes risk of parameter sensitivity that is a result of parameter misspecifi-cation. Type 0 arises due to a lack of information in liquidly traded market instruments.

2. Type 1 is the inability of the model to recover a complete set of market obser-vations. Type 1 is referred to as calibration error - the inability to recover the market observations, irrespective of the model’s dynamics.

3. Type 2 is the risk incurred as a result of recalibration error. This error is a con-tradiction of the model assumptions (due to the lack of parameter stability). Over time, the model needs to be recalibrated in order to have an accurate fit.

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2.2 Model Risk Quantification 4

The frequent calibrations are a violation of the model’s assumption of fixed parameters.

4. Type 3 describes the contradiction of the model assumptions with its em-pirical dynamics. This is as a result of a misspecification of the stochastic dynamics.

Using this framework will enable us to discern the factors affecting model risk in the quantification of model risk in Value-at-Risk models.

2.2 Model Risk Quantification

We begin by defining theKullback and Leibler (1951) divergence and thereafter

give a brief overview of model risk according toGlasserman and Xu(2014).

Definition 2.1. The Kullback-Leibler (KL) divergence between the nominal

distri-bution dQ, and the alternative distridistri-bution dP , is defined as, D(Q||P ) = Z logdQ dP dQ.

Definition 2.2. The KL divergence has the following properties:

i. D(Q||P ) ≥ 0.

ii. D(Q||P ) = 0 if and only if dQ = dP almost everywhere. iii. D(Q||P ) 6= D(P ||Q).

Note that D(Q||P ) is not a metric. Although it measures the ”distance” be-tween two disparate distributions, D(Q||P ) is not symmetric. That is, property iii

of Definition2.2does not satisfy property iii of a metric in DefinitionA.1. However,

properties i and ii of the KL divergence satisfies properties i and ii of a metric

(de-fined in DefinitionA.1). Despite this, the KL divergence has attractive properties

inherent to a metric and can be considered as a pseudo-metric. It should be noted thatGlasserman and Xu(2014) do not view dQ and dP as symmetric distributions and we consider the possibility of dP as a more suitable distribution although dQ might be the favoured distribution.

Glasserman and Xu(2014) use the KL divergence in order to quantify the ”tance” between two distributions. This requires the selection of a nominal dis-tribution, to which the alternative distribution will be compared. This pseudo-metric measures the information gained that would make the alternative distribu-tion preferable.

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2.2 Model Risk Quantification 5

Model risk, defined byGlasserman and Xu(2014), is classified as the worst-case

error for a risk measure V (X) on the random variable X. In our context, the risk measure V (X) denotes Value-at-Risk (VaR) on a portfolio of European options. In order to quantify model risk, consider alternative models that are a ”distance” of

ηfrom the nominal model, where η is termed the relative entropy budget. The

al-ternative models are described by the set Pηof Radon-Nikodym derivatives m(X)

such that,

E[m(X) log m(X)] < η.

Suppose the Radon-Nikodym derivatives are given by m = dQ_dP, which assumes,

without loss of generality, that dP > 0. Further suppose that QX, PX have densities

fand ˜f respectively. The expectation of V (X) under the alternative distribution dP

is, ˜ E[V (X)] = E[m(X)V (X)] = Z m(x)V (x)f (x)dx = Z V (x) ˜f (x)dx. (2.1)

It follows that the worst-case lower and upper bounds are given by, inf

m∈Pη

E[m(X)V (X)] and sup

m∈Pη

E[m(X)V (X)], (2.2)

respectively. In order to determine the maximisation problem in Equation (2.2),

the primal is given by1,

− inf

m∈Pη

E[−m(X)V (X)] subject to E[m(X) log(m(X))] − η ≤ 0. (2.3)

In this context, the function that is to be minimised, g(m) = E[−m(X)V (X)], is linear in m and thus convex. The constraint, h(m) = E[m(X) log(m(X)) − η]

is also convex. Thus the minimisation approach to equation (2.3) is given by the

following, inf δ>0supm E m(X)V (X) − 1 δ m(X) log m(X) − η . (2.4)

Glasserman and Xu (2014) state that, with the Lagrangian multiplier δ being strictly positive, the worst possible risk within the distance η will be given by,

inf δ>0_m∈Psup_ηE m(X)V (X) − 1 δ m(X) log m(X) − η , with,

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2.3 Robust Monte Carlo 6

m(X) = exp(δV (X))

E[exp(δV (X))].

Similarly, the lower bound is solved with δ being strictly negative. The worst-case model error is characterised by an exponential change of measure which is defined in terms of the risk measure V (X) and the parameter δ.

2.3 Robust Monte Carlo

Glasserman and Xu(2014) describe a robust Monte Carlo approach that estimates model risk.

Consider the problem supm∈PηE[m(X)V (X)], where the dual optimisation is

described by Equation (2.4). The inner supremum has a solution of the form,

mδ(X) =

eδV (X)

E[eδV (X)]

.

Suppose independent replications X1, X2, · · · , XN of the random element X is

generated. The Monte Carlo estimate of E[V (X)] is determined by, 1 N N X i=1 V (Xi).

For a given δ and the Radon-Nikodym derivative mδ ∝ eδV (X)_{, the expectation}

of V (X) can be estimated using the change of measure mδ, determined by Xiunder

the nominal measure. The estimator, ˜ E[V (X)] = E[m(X)V (X)] = PN i=1V (Xi)eδV (Xi) PN i=1eδV (Xi) , (2.5)

converges to E[mδ(X)V (X)]as N → ∞. With the same generated replications,

the Radon-Nikodym derivative can be estimated by, ˆ mδ,i = eδV (Xi) 1 N PN j=1eδV (Xj) , i = 1, · · · , N. (2.6)

Then, estimation of the relative entropy of mδis given by,

ˆ η(δ) = 1 N N X i=1 ˆ

mδ,ilog ˆmδ,i. (2.7)

By varying δ, the risk measure V (X), under various exponential change of mea-sures mδ, can be estimated and an optimal δ can be selected where,

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Chapter 3 Option Pricing Models

In this chapter we give a brief overview of the models we considered for option

pricing: the Black-Scholes model (Black and Scholes(1973)), which assumes a

con-stant volatility, the Heston Stochastic Volatility model (Heston(1993)), which

re-produces the volatility smile, and the Variance-Gamma model (Madan et al.(1998)),

which assumes information is known at random times. Options are written on the ALSI (All Share Index) Futures index.

3.1 The Black-Scholes model

The Black-Scholes model for European options assumes that the stock price pro-cess follows Geometric Brownian Motion (GBM). Assume a risk-neutral setting

(Ω, F , Q), with market information (Ft)t≥0 at time t. The risk-neutral stock price

process dynamics are,

dSt= rStdt + σBSStdWt, (3.1)

where r is the risk-free rate of return, σBS > 0 is the constant volatility of the

underlying St, and Wtis the Q-Brownian motion. The corresponding Black-Scholes

(BS) price for a European call option is,

cBS_t = StN (d+) − Ke−rτN (d−), (3.2)

where N (·) denotes the cumulative standard normal distribution, K is the strike

price, (St)t≥0is the stock price process, the expiry date is T and the time-to-maturity

is τ = T − t. d+= log(St/K) + (r + 0.5σBS2 )τ σBS√τ , d−= d+− σBS √ τ . (3.3)

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3.2 The Heston Stochastic Volatility model 8

3.2 The Heston Stochastic Volatility model

The Heston Stochastic Volatility model (Heston(1993)), allows the volatility of the

stock price process to change stochastically. Contrary to the Black-Scholes model, the volatility of the underlying is described by a diffusion process. This character-istic allows the Heston model to reproduce the volatility smile.

In a risk-neutral setting (Ω, F, Q), with the filtration (Ft)t≥0, which represents

the market information at time t, the stock price process is given by,

dSt= rStdt + √ vtStdWt dvt= κ(θH − vt)dt + σH √ vtdBt, (3.4)

where r is the risk-free rate of return, the instantaneous variance vtwith v0≥ 0,

the mean-reversion speed κ > 0, the mean-reversion level θH > 0and volatility

of the volatility process σH > 0. The instantaneous volatility, (vt)t≥0, follows the

Cox et al.(1985) interest rate term structure. Wtand Btare dependent Q- standard Brownian motions such that dhWt, Bti = ρdt, where ρ ∈ [−1, 1] is the correlation

between the Brownian motions. The set ΘHeston = {v0, θH, ρ, κ, σH} is considered

the set of Heston models parameters.

If Feller’s condition, defined as 2κθH > σ2_H, is met then the volatility process

will be strictly positive.

Define Xt= log St. The dynamics of the log-price process Xtfollows from Ito’s

formula as, dXt= r −vt 2 dt +√vtdWt. (3.5)

The characteristic function provided byAlbrecher et al.(2007) is given by,

φHeston(u, τ ) = exp[C(τ, u) + D(τ, u) · v0+ iux0], (3.6)

where X0 = x0 is the known log-price at time t = 0, and,

C(τ, u) = iurτ + κθH σ_H2 (κ − iuρσH − d)τ − 2 log 1 − ge−d·τ 1 − g , D(τ, u) = κ − iuρσH − d σ_H2 · 1 − e−d·τ 1 − g · e−d·τ, g = κ − iuρσH − d κ − iuρσH + d, d = q (iuρσH− κ)2_{+ σ}2 H(iu + u2). (3.7)

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3.3 The Variance-Gamma Model 9

The price of a European call option under these dynamics (Gil-Pelaez(1951)) is

given by, cH(St, K, τ ) = S0P1− Ke−rτP2, Where P1= 1 2 + 1 π Z ∞ 0 Re e −iu ln(K)_{φHeston(u − i, τ )}

iuφHeston(−i, τ )

du, P2= 1 2 + 1 π Z ∞ 0 Re e −iu ln(K)_φ Heston(u, τ ) iu du (3.8)

where K is the strike price, r is the risk-free rate of return, τ is the

time-to-maturity and φHeston(·, τ )is the characteristic function of the log-price at the time

of expiry, T .

3.3 The Variance-Gamma Model

The Variance-Gamma model (VG)(Madan et al.(1998)) is obtained via a Brownian

motion with constant drift and volatility. The VG model assumes that market infor-mation is obtained at random times. This random time change is modelled using a gamma process. Each unit of time is given by an independent random variable that is driven by a gamma density with a unit mean and positive variance, ν.

In order to obtain the European The VG process is obtained by evaluating the

Brownian motion b(t; θV G, σV G)given by,

db(t; θV G, σV G) = θV Gdt + σV GdWt,

with the drift θV G, volatility σV G at a random time modelled by the gamma

process and Wtis a standard Brownian motion.

The VG process, X(t; σV G, ν, θV G), is obtained in terms of a Brownian motion

with drift b(t; θV G, σV G)and gamma process with a unit mean rate y(t; 1, ν) given

as:

X(t; σV G, ν, θV G) = b(y(t; 1, ν); θV G, σV G). (3.9)

The risk-neutral stock prices are given by,

St= S0exp((r + ω)t + X(t; σV G, ν, θV G)),

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3.3 The Variance-Gamma Model 10 φV G(u, τ ) = exp[iu(log S0+ (θV G+ ω − 1 2σ 2 V G)τ )]× φXt(u, τ ) exp h −1 2(uσV G) 2_τi_, (3.10)

where ω = _ν1ln(1 − θV Gν − 0.5σV G2 ν)and φXtis given by,

φXt(u, t) = 1 1 − iuνθV G+ 0.5ν(uσV G)2 !t ν (3.11)

Equation (3.10) can be used to derive the price of a European call option,

cV G(St, K, τ ) = S0P1− Ke−rτP2, Where P1= 1 2 + 1 π Z ∞ 0 Re e −iu ln(K)_{φV G(u − i, τ )} iuφV G(−i, τ ) du, P2= 1 2 + 1 π Z ∞ 0 Re e −iu ln(K)_φ V G(u, τ ) iu du (3.12)

where λ > 0 is a damping parameter that ensures integrability. All the other

input parameters are as previously defined. ΘV G = {σV G, ν, θV G} is considered

the set of VG model parameters.

The corresponding European put option price for the Black-Scholes, Heston and the VG models are obtained via put-call parity.

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Chapter 4 Model Calibration

This chapter details the implementation of the Black-Scholes, Heston and VG SAFEX (South African Futures Exchange) European options written on the ALSI (All Share Index) Futures contracts. This is preceded by a description of the data.

Calibration of the option pricing models entails solving a constrained non-linear optimisation problem in order to determine the risk-neutral, Q-measure, model pa-rameters Θ which best fits the market data. Optimisation requires the use of an ob-jective function (or error measure) which can be defined in various ways, depend-ing on the calibration instruments used. Calibration instruments refer to quoted option prices and implied volatilities.

For the sake of consistency, the same objective function is used for the Black-Scholes, Heston and VG calibration. The relative squared price error (RSPE)

objec-tive function (Escobar and Gschnaidtner(2016)) is defined as,

RSPE = ND X i=1 cM odel i − cM arketi cM arket i 2 , (4.1)

where ND is the number of instruments used in the calibration.

Calibration on the various option pricing models is performed using MATLAB’s built-in fmincon optimisation function together with the sqp algorithm. Due to time and resource constraints, local minima were found.

4.1 Data

The data regarding SAFEX European options on ALSI futures contracts in the mar-ket is quoted in terms of implied volatilities. The ALSI index is comprised of 40 of the largest listed companies on the JSE, weighted by market capitalisation. Strikes range from 80% to 120% moneyness with increments of 5%. The maturities are {30, 90, 180, 270} days-to-maturity. The data ranges from 04 November 2008 to 29 June 2018. Due to the risk of using data that might be outdated, the data was

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4.2 Black-Scholes Calibration 12

truncated to 1500 days, spanning from 25 June 2012 to 29 June 2018. Calibration of

σBS, ΘHestonand ΘV Gis obtained using 20 days time series data from 25 June 2012.

4.2 Black-Scholes Calibration

The Black-Scholes model requires calibration of a single parameter, viz. the con-stant volatility σBS. The analytical Black-Scholes option price is given by equation

(3.2). The Black-Scholes calibration results are given in Table4.1. Figure4.1shows

the calibrated Black-Scholes call option prices compared to the market implied call option prices.

Days-to-maturity

Parameters 30 90 180 270

σBS 20.50% 20.50% 21.35% 21.57%

RSPE 0.0032 0.0181 0.0345 0.0443

Table 4.1:Black-Scholes (BS) calibration results.

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Moneyness 0 1 2 3 4 5

Call Option Prices

104 30 days to maturity

BS

Market Implied Call Option

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Moneyness 0 1 2 3 4 5 6

Call Option Prices

BS

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Moneyness 0 1 2 3 4 5 6

Call Option Prices

BS

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Moneyness 0 1 2 3 4 5 6

Call Option Prices

BS

Figure 4.1:Comparison of Black-Scholes and market implied call option prices

across various days-to-maturities.

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compar-4.3 Heston Model Calibration 13

ison to the market implied call prices. It can be seen that for options with longer time-to-maturity, the Black-Scholes assumption of a constant volatility does not re-cover the market implied call prices. This effect can be seen on options with 270 days-to-maturity.

4.3 Heston Model Calibration

Calibration of the Heston model requires determination of the five model

param-eters, ΘHeston = {v0, θH, ρ, κ, σH}. Semi-analytical formulae are available for the

computation of the European call option prices. These are,

cH_t (St, τ ) = StP1− Ke−rτP2 with P1 = 1 2+ 1 π N X n=1 Re he−iunkφHeston(un− i, τ ) iunφHeston(−i, τ ) i ∆u P2 = 1 2+ 1 π N X n=1 Rehe −iunk_φ Heston(un, τ ) iun i ∆u, (4.2)

where k = ln(K), ∆u = umax

N and un = (n −

1

2)∆u. The integration limits are

truncated to the interval [0, umax], where the upper integration limit is umax = 30

and the number of quadrature steps is N = 100.

The following specifications recommended byEscobar and Gschnaidtner(2016)

were implemented,

Function Tolerance: 1e-10

Maximum Function Iterations: 10000

Maximum Iterations: 50000

With the following lower bound (lb) and upper bound (ub) constraints,

lb: {1e-5, 1e-5, -0.99, 1e-5, 1e-5}

ub: {3, 3, 0.99, 20, 3}

The Heston calibration results are given in Table4.2.

It should be noted that Feller’s condition was not enforced during the calibra-tion of the Heston model as either an initial guess or as a constraint. This is evident

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4.3 Heston Model Calibration 14 Days-to-maturity Parameters 30 90 180 270 v0 0.0491 0.0847 0.2338 0.2838 θH 0.0082 0.0335 0.0214 0.0278 ρ 0.988 0.7331 0.7434 0.7638 κ 1.1929 19.5812 15.1437 14.4774 σH 0.1742 0.9676 1.4246 1.8722

RSPE 0.001210689 0.00011962 1.29e-05 1.02e-06

Table 4.2:Heston model calibration results

in the resultant calibrated parameters as Feller’s condition is not met for all

days-to-maturity. According toCui et al.(2017), relaxing Feller’s condition is not always

detrimental to modelling and may be beneficial for derivative pricing.

Figure4.2 shows the Heston implied volatility smile compared to the market

implied volatility smile and the Black-Scholes constant volatility. Figure4.3shows

the Heston call option prices compared to the market implied call option prices.

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Moneyness 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 Implied Volatility 30 days to maturity Black-Scholes Implied Volatility Heston 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Moneyness 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 Implied Volatility 90 days to maturity Black-Scholes Implied Volatility Heston 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Moneyness 0.16 0.18 0.2 0.22 0.24 0.26 0.28 Implied Volatility 180 days to maturity Black-Scholes Implied Volatility Heston 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Moneyness 0.16 0.18 0.2 0.22 0.24 0.26 0.28 Implied Volatility 270 days to maturity Black-Scholes Implied Volatility Heston

Figure 4.2:Market implied volatility smile, Heston volatility smile and BS constant

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4.4 Variance-Gamma Calibration 15 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Moneyness -1 0 1 2 3 4 5

Call Option Prices

Heston

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Moneyness 0 1 2 3 4 5

Call Option Prices

Heston

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Moneyness 0 1 2 3 4 5 6

Call Option Prices

Heston

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Moneyness 0 1 2 3 4 5 6

Call Option Prices

Heston

Figure 4.3:Comparison of Heston and market implied call option prices across

var-ious days-to-maturities.

Recall that the Heston model is calibrated using market call option prices. The calibrated Heston call prices compared to the market call prices is shown in

Fig-ure 4.3. The corresponding Heston volatility smile is inferred from the Heston

call prices. Figure4.2 shows the Heston volatility smile compared to the market

volatility smile. The deviation between the Heston volatility smile and the market volatility smile is due to the calibration on call prices.

4.4 Variance-Gamma Calibration

Calibration of the VG model entails determining the VG model parameters, ΘV G=

{σV G, ν, θV G}. The VG European option price is given by,

cV G_t (St, τ ) = StP1− Ke−rτP2 with P1 = 1 2+ 1 π N X n=1 Re he−iunkφ_{V G}(u_n− i, τ ) iunφV G(−i, τ ) i ∆u P2 = 1 2+ 1 π N X n=1 Re he−iunkφV G(un, τ ) iun i ∆u, (4.3)

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4.4 Variance-Gamma Calibration 16

where φV G(u, t)is given by equation (3.10), k = ln(K), ∆u = umax

N and un =

(n−1₂)∆u. The integration limits are also truncated to the interval [0, umax]with the

upper integration limit umax= 30, and the number of quadrature steps N = 100.

The VG calibration results are given in Table4.3. Note that with the exception

of τ = 30 days-to-maturity, ν increases with an increase in τ . The exception of ν when τ = 30 days-to-maturity can be attributed to the nature of the data and ALSI market dynamics. Days-to-maturity Parameters 30 90 180 270 σV G 0.122 0.103 0.096 0.089 ν 1.881 0.159 0.447 0.695 θV G 0.103 0.378 0.254 0.212

RSPE 0.01 0.001 6.33e-03 4.51e-05

Table 4.3:VG calibrated parameters

Figure4.4 shows the VG implied volatility smile compared to the market

im-plied volatility smile and the Black-Scholes constant volatility. Figure4.5shows the

VG call option prices compared to the market implied call option prices.

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Moneyness 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 Implied Volatility 30 days to maturity Black-Scholes Implied Volatility VG 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Moneyness 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 Implied Volatility 90 days to maturity Black-Scholes Implied Volatility VG 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Moneyness 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 Implied Volatility 180 days to maturity Black-Scholes Implied Volatility VG 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Moneyness 0.16 0.18 0.2 0.22 0.24 0.26 0.28 Implied Volatility 270 days to maturity Black-Scholes Implied Volatility VG

Figure 4.4:Market implied volatility smile, VG volatility smile and BS constant

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4.4 Variance-Gamma Calibration 17 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Moneyness -1 0 1 2 3 4 5

Call Option Prices

VG

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Moneyness 0 1 2 3 4 5

Call Option Prices

VG

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Moneyness 0 1 2 3 4 5 6

Call Option Prices

VG

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Moneyness 0 1 2 3 4 5 6

Call Option Prices

VG

Figure 4.5:Comparison of VG and market implied call option prices across various

days-to-maturities.

The VG model is calibrated using market call option prices. The calibrated VG

call prices compared to the market call prices is shown in Figure4.5. The

corre-sponding Heston volatility smile is inferred from the calibration of VG prices.

Fig-ure4.4shows the VG volatility smile compared to the market volatility smile. The

deviation between the VG volatility smile and the market volatility smile is due to the calibration on call prices.

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Chapter 5 Static Hedging and VaR

This chapter begins with a discussion of path simulation under the real-world mea-sure, P (via a Bootstrap method) and the risk-neutral meamea-sure, Q. This is followed by a discussion on static delta-hedging of European options described in

Chap-ter3. This is followed by a discussion of Value-at-Risk (VaR) of a portfolio under

the real-world measure, P, and risk-neutral measure, Q. The real-world measure, P, assumes the use of historical data whilst the risk-neutral measure, Q, relies on risk-neutral simulation.

5.1 Path Simulation

The paths of the underlying are simulated via a Bootstrap method and a risk-neutral simulation. The Bootstrap method is the real-world measure, P, which relies on historical data. Risk-neutral simulation is the risk-neutral measure, Q, where the paths are determined in a model-dependent manner.

5.1.1 Bootstrap Method

In order to simulate the sample path via bootstrap, suppose we have a historical time series of spot prices for the underlying. The daily log returns for the

underly-ing can be obtained and denoted as R = [R1, · · · , RN]. That is,

Ri= log Sti Sti−1 .

At time t0, we select the latest available underlying spot price as St0. At the

hedging date ti∈ [t1, · · · , th]we randomly select Ri ∈ R. Thus,

Sti = Sti−1e

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5.1 Path Simulation 19

5.1.2 Risk-Neutral simulation

The risk-neutral path simulation of the underlying is model dependent. In this con-text, the underlying follows the Black-Scholes, Heston and Variance Gamma model

assumptions respectively. We begin by defining theMilstein(1994) approximation

scheme for generating paths.

Definition 5.1. Milstein Approximation:

Suppose a stochastic process Y defined by,

dYt= a(Yt)dt + b(Yt)dWt, Y0is a constant,

where a, b ∈ C2(R). TheMilstein(1994) approximation is a sequence of random

variates ˆY0, · · · , ˆYN at times tiwith ∆ti = ti− ti−1for i ≥ 0 given by,

ˆ Yti =

(

Yt0 ti= t0,

ˆ

Yti−1+ a( ˆYti−1)∆ti+ b( ˆYti−1)∆Wti+

1 2b( ˆYti−1) db dx( ˆYti−1)((∆Wti) 2_{− ∆t} i) otherwise,

with Wtas the standard Brownian motion and ∆Wti = Wti− Wti−1.

FollowingGlasserman(2013), the risk-neutral sample path for the underlying

under BS assumptions using Milstein approximation is given by,

Sti = Sti−1+ rSti−1∆ti+ σBSSti−1

p ∆tiZi+ 1 2σBSSti−1(Z 2 i − 1)∆ti,

where ∆ti = ti− ti−1, r is the risk-free rate and Zi is a random number generated

from the standard normal distribution. Similarly using the Milstein approximation,

the Heston model sample path as suggested byRouah (2013) and Hirsa(2016) is

given by,

Sti = Sti−1e

(r−1₂v_ti−1)∆ti+

√

v_ti−1∆tiZs,i

vti = max vti−1+ κ(θH − vti−1)∆ti+ σH

q vti−1∆tiZv,i+ 1 4σ 2 H(Zv,i2 − 1)∆ti, 0,

with v0and S0 at hedging date t0. The standard normal random variables Zs,i

and Zv,i have correlation ρ.

In the case of the Variance Gamma,Fu et al.(2007) suggests that the sample path

for an underlying is given by,

Xti = Xti−1+ θV G∆Gi+ σV G

p ∆GiZi Sti = Sti−1e

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5.2 Static Hedging 20

where ∆Gi is a random variable generated from the Gamma distribution with

shape parameter ∆ti

ν and and scale parameter ν, whilst Zi is a random variable

generated from the standard normal distribution and ω =_ν1ln(1−θV Gν −0.5σ_{V G}2 ν).

Figure5.1below shows the distributions of the underlying, St10, after 10 days,

assuming that one can only observe the underlying at the end of the day. The distributions of the underlying are shown for the Historical Bootstrap, BS, Heston and VG models. The distributions were fitted using MATLAB’s built-in ksdensity function. 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 6.2 Price 105 0 0.2 0.4 0.6 0.8 1 1.2 10 -4 Historical Bootstrap Black-Scholes Heston VG

Figure 5.1:Distribution of St10 under Historical Bootstrap, Black-Scholes, Heston

and Variance Gamma assumptions after 10 days.

Figure5.1shows that the Black-Scholes and VG underlying price distribution at

time t10is similar to the Historical Bootstrap distribution. The Heston distribution

has a higher peak and longer tails.

5.2 Static Hedging

The delta of an option is regarded as the sensitivity of an option value to the change

in the underlying price. The delta of a European call option, denoted ∆C, is defined

as,

∆C = ∂c(S0, τ )

∂S0 . (5.1)

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5.2 Static Hedging 21

∆P =

∂p(S0, τ ) ∂S0

. (5.2)

In the case of the Black-Scholes model, the delta of European call and put op-tions have analytical soluop-tions given by,

∆C = N (d+) and ∆P = −N (−d+), (5.3)

respectively where N (·) is the cumulative distribution function of the standard

normal distribution and d+is given by Equation3.3.

Similarly the delta of the Heston and VG model is found by taking the

deriva-tive of the semi-analytical formulae, given by Equations4.2and4.3, with respect to

S0.

The delta of the European call and put Heston options is given by,

∆C = ∂c H_(St) ∂S0 = P1+ S0 ∂P1 ∂S0 − Ke −rτ∂P2 ∂S0 ∆P = ∂p H_(S t) ∂S0 = S0 ∂P1 ∂S0 − (1 − P1) − Ke −rτ∂P2 ∂S0. (5.4) respectively where, P1= 1 2 + 1 π N X n=1 Re e −iunln(K)_φHeston(un_{− i, τ )} iunφHeston(−i, τ ) ∆u S0 ∂P1 ∂S0 = 1 π N X n=1 Re e−iunln(K)φHeston(un− i, τ ) φHeston(−i, τ ) ∆u ∂P2 ∂S0 = 1 π N X n=1 Re e −iunln(K) iun φHeston(un, τ ) ∆u.

Similarly the delta of the European call and put options under the VG model is given by, ∆C = ∂cV G(St) ∂S0 = P1+ S0 ∂P1 ∂S0 − Ke −rτ∂P2 ∂S0 ∆P = ∂pV G(St) ∂S0 = S0 ∂P1 ∂S0 − (1 − P₁) − Ke−rτ∂P2 ∂S0 . (5.5) respectively where,

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5.2 Static Hedging 22 P1 = 1 2 + 1 π N X n=1 Re e −iunln(K)_φ V G(un− i, τ ) iunφV G(−i, τ ) ∆u S0 ∂P1 ∂S0 = 1 π N X n=1 Re e−iunln(K)φV G(un− i, τ ) φV G(−i, τ ) ∆u ∂P2 ∂S0 = 1 π N X n=1 Re e −iunln(K) iun φV G(un, τ ) ∆u.

Figure 5.2 below illustrates BS, Heston and VG delta at time t0 for the data

described in Section4.1. 0 300 0.2 0.4 1.2 0.6 200 BS Delta 1.1 days-to-maturity 0.8 Moneyness 1 1 100 0.9 0 0.8 -0.2 300 0 0.2 0.4 1.2 200 0.6 Heston Delta 1.1 days-to-maturity 0.8 Moneyness 1 1 100 0.9 0 0.8 -0.2 300 0 0.2 0.4 1.2 200 0.6 VG Delta 1.1 days-to-maturity 0.8 Moneyness 1 1 100 0.9 0 0.8

Figure 5.2: ∆C for BS, Heston and VG at time t0

The procedure for delta hedging a European call and put option is described by the following algorithm. It should be noted that this approach assumes no restric-tions on the ability to long or short a financial instrument.

Static Delta Hedging Algorithm European call option:

1. We set up a portfolio by longing the European call option and shorting ∆C

units of the underlying. We denote the P&L of a portfolio at time t as Πtand

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5.3 VaR 23

Πt0 = c(St0, τ ) − ∆CSt0.

2. At time n, the Profit and Loss (P&L) is given by, Πtn = Πt0e

r∆tn_{+ ∆CSt}

n− Φ(Stn),

where Φ(Stn) denotes the value of the option at time t. That is, Φ(ST) =

max(ST − K, 0) if the option is hedged until maturity.

European put option:

1. Similarly, we set up a portfolio by shorting the European put option and

shorting ∆P units of the underlying. Therefore the initial residual P&L of

the portfolio is given by,

Πt0 = −|∆P|St0− p(St0, τ ).

2. At time n, the P&L of the put option is given by, Πtn = Πt0e

r∆t0_{+ |∆P}_|S

tn+ Φ(Stn).

It should be noted that the P&L of the European options are determined using

the calibrated parameters specified in Chapter4.

5.3 VaR

VaR is defined as the loss that is not expected to be exceeded with a certain signifi-cance level over a period of time. VaR was initially developed by RiskMetrics at JP Morgan and later became adopted by financial practitioners as a tool to measure

risk (seeGuldimann et al.(1995)).

VaR has two parameters viz. α, the significance level (or the confidence level 1 − α) and the risk horizon, h, which is the period of time over which VaR will be calculated. The risk horizon is the period over which one is exposed to a position. This means that the more illiquid the risk, the longer the period over which the risk should be assessed. That is, positions which cannot be closed or hedged quickly should be assessed over longer risk horizons.

Recall that the Profit and Loss (P&L) of a portfolio at time this denoted by Πth.

Πth is calculated by static hedging of European options. Let VaR be denoted as xh.

In order to estimate VaR at time th, we need to find the α−quantile xh such that

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5.3 VaR 24

VaRh = − inf{xh ∈ R | P[Πth ≤ xh] ≥ α}

We quantify the model risk inherent in the Historical VaR and Monte Carlo (MC) VaR. Historical VaR is a real-world measure, P and calculated via a bootstrap method on historical data whilst MC VaR is a risk-neutral Q−measure calculated by simulating the risk-neutral paths for the underlying.

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Chapter 6 Results

Recall that this dissertation quantifies the model risk in VaR of certain portfolios

by using the methodology described byGlasserman and Xu(2014). This

method-ology is used in order to compare the real-world P- measure Historical VaR to the risk-neutral Q-measure Monte Carlo VaR through the use of the parameter δ. By analysing various portfolios under the change of measure, we search for δ without solving for one. This chapter begins by analysing the Profit and Loss (P&L) dis-tribution model risk in various portfolios based on Black-Scholes, Heston and VG models. This is followed by a discussion of the model risk in VaR under the various portfolios. The portfolios are observed over a risk horizon period of 10 days with a rate of return, r, of 7% and VaR is determined for a 95% confidence level.

The following portfolios were considered:

1. One At-The-Money (ATM) call option for each time-to-maturity.

2. Long-only portfolio with three call options with moneyness of 95%, 100% and

105%respectively.

3. Collar portfolio with a long position in an out-the-money put with a strike of

85%moneyness, short position in an out-the-money call with a strike of 115%

moneyness and a long position in the ALSI future itself.

6.1 Profit and Loss Distributions

Recall from Chapter2that in order to quantify model risk, an exponential change

of measure is required. The exponential change of measure is characterised by the parameter δ and the risk measure V (X). In our context, X refers to the P&L of the

portfolio and V (X) is VaR of the portfolio. E[mδ(X)V (X)]refers to the worst-case

value of V (X) implied by the parameter δ. In order to analyse the effect of the expo-nential change of measure on the MC P&L and MC VaR, various values of the pa-rameter δ were arbitrarily chosen. The chosen values were δ = [−1.5, −0.5, 0.5, 2].

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6.1 Profit and Loss Distributions 26

Figures6.1 - 6.3 illustrate the effect of the change of measure in order to

per-turb the MC P&L for the three portfolios on options with 90 days-to-maturity. The Historical and the perturbed P&L distributions for each portfolio were fitted using MATLAB’s built-in ksdensity function.

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 P&L 104 0 0.2 0.4 0.6 0.8 1 1.2 10 -3 BS: = 90days Nominal = -1.5 = -0.5 = 0.5 = 2 -6 -5 -4 -3 -2 -1 0 1 P&L 104 0 0.2 0.4 0.6 0.8 1 1.2 1.4 10 -3 Heston: = 90days Nominal = -1.5 = -0.5 = 0.5 = 2 -5 -4 -3 -2 -1 0 1 P&L 104 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 10 -3 VG: = 90days Nominal = -1.5 = -0.5 = 0.5 = 2

Figure 6.1:ATM call P&L distributions for BS, Heston and VG 90 with

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6.1 Profit and Loss Distributions 27 -7 -6 -5 -4 -3 -2 -1 0 1 P&L 104 0 0.5 1 1.5 2 2.5 3 3.5 4 10 -4 BS: = 90days Nominal = -1.5 = -0.5 = 0.5 = 2 -7 -6 -5 -4 -3 -2 -1 0 1 P&L 104 0 0.5 1 1.5 2 2.5 3 3.5 4 10 -4 Heston: = 90days Nominal = -1.5 = -0.5 = 0.5 = 2 -8 -7 -6 -5 -4 -3 -2 -1 0 1 P&L 104 0 0.5 1 1.5 2 2.5 3 3.5 4 10 -4 VG: = 90days Nominal = -1.5 = -0.5 = 0.5 = 2

Figure 6.2:Long-only portfolio P&L distributions for BS, Heston and VG with 90

days-to-maturity. -1 0 1 2 3 4 5 P&L 104 0 1 2 3 4 5 6 10 -4 BS: = 90days Nominal = -1.5 = -0.5 = 0.5 = 2 -1 0 1 2 3 4 5 P&L 104 0 1 2 3 4 5 6 10 -4 Heston: = 90days Nominal = -1.5 = -0.5 = 0.5 = 2 -1 0 1 2 3 4 5 6 7 P&L 104 0 0.5 1 1.5 2 2.5 3 3.5 4 10 -4 VG: = 90days Nominal = -1.5 = -0.5 = 0.5 = 2

Figure 6.3:Collar portfolio P&L distributions for BS, Heston and VG models with

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6.2 VaR model risk 28

Note that the parameter δ affects some properties of the unperturbed MC P&L distribution. The δ shifts the distributions for each portfolios. The change of mea-sure on the respective P&L distributions changes the moments of the distributions. If δ can be interpreted as a pseudo-risk aversion measure, then δ < 0 can be in-terpreted as some level of risk aversion and the worst-case shifts the profitability towards a loss. On the other hand, δ > 0 can be interpreted as a level of risk-taking and shifts the P&L in the opposite direction.

It should be noted that δ < 0 does not have a symmetric effect on the P&L distribution as δ > 0. Similarly, the effect of δ varies with each portfolio.

The effect of the exponential change of measure for portfolios on options with

30, 180 and 270 days-to-maturity is presented in FiguresB.1-B.3,B.4-B.6andB.7

-B.9respectively.

6.2 VaR model risk

Recall from Section5.3that Historical and Monte Carlo VaR was obtained by

boot-strapping and simulating the risk-neutral sample paths respectively. Table6.1presents

the 95% VaR over a 10 day period, for the various portfolios consisting of options with 90 days-to-maturity for Black-Scholes, Heston and VG, respectively.

τ = 90 BS Heston VG

ATM Historical 3 429.00 3 432.41 3 316.15

Monte Carlo 3 712.18 3 130.73 593.97

Long Only Historical 9 530.04 9 492.48 8 963.50

Monte Carlo 10 428.54 8 612.07 2 007.54

Collar Historical 2 171.70 1 737.83 3 339.13

Monte Carlo 2 171.50 1 739.41 3 344.06

Table 6.1: 10-day VaR of portfolios with τ = 90 days-to-maturity

From Table6.1above, Black-Scholes and Heston Monte Carlo VaR is

commen-surate with its Historical VaR. With the exception of the collar portfolio, there is a disparity between the VG Monte Carlo VaR and the Historical VaR. VaR of portfo-lios with 30, 180, 270 days-to-maturity exhibit similar results and are given in Tables

B.1-B.3.

Figure6.4 illustrates the effect on VaR on an ATM portfolio under the change

of measure and the difference between the nominal distribution and the perturbed VaR. The effect of the exponential change of measure can be seen in the values of

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the difference at each day-to-maturity. The greatest difference across the stochastic models occurs for options with 30 days-to-maturity, for each δ. Under each stochas-tic model and for each day-to-maturity, δ = 0.5 gives the least difference.

Similarly, Figures 6.5 and 6.6 illustrate the effect of the parameter δ on VaR,

based on a long-only and collar portfolio respectively, including the differences between the nominal VaR and perturbed VaR. For a long position in an ATM call and the long-only portfolio, δ = 0.5 provides a commensurate perturbed MC VaR whilst δ = −0.5 provides a commensurate perturbed collar MC VaR.

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6.2 VaR model risk 30 0 50 100 150 200 250 300 Days to Maturity 0 0.5 1 1.5 2 2.5 3 3.5 95% VaR 104 =-1.5 =-0.5 =0.5 =2.5 Nominal 0 50 100 150 200 250 300 Days to Maturity -0.5 0 0.5 1 1.5 2 2.5 3 Difference 104 =-1.5 =-0.5 =0.5 =2.5 (a)BS: 95% VaR 0 50 100 150 200 250 300 Days to Maturity 0 0.5 1 1.5 2 2.5 3 3.5 4 95% VaR 104 =-1.5 =-0.5 =0.5 =2.5 Nominal 0 50 100 150 200 250 300 Days to Maturity -0.5 0 0.5 1 1.5 2 2.5 3 Difference 104 =-1.5 =-0.5 =0.5 =2.5 (b)Heston 95% VaR 0 50 100 150 200 250 300 Days to Maturity 0 0.5 1 1.5 2 2.5 3 3.5 4 95% VaR 104 =-1.5 =-0.5 =0.5 =2.5 Nominal 0 50 100 150 200 250 300 Days to Maturity -0.5 0 0.5 1 1.5 2 2.5 3 Difference 104 =-1.5 =-0.5 =0.5 =2.5 (c)VG: 95% VaR

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It is worth noting that the change of measure has an effect on VaR dependent on the portfolio, time-to-maturity and parameter δ.

Recall from Section 2.1 the model risk classification which provides a

frame-work in which model risk in the various portfolios can be attributed to a variety of factors. These are:

• the sparse available data. That is, the ALSI options data obtained only con-sists of [30, 90, 180, 270] days-to-maturity

• the initial assumptions regarding the assumptions about the underlying’s stochastic behaviour

• truncating the integration limits during valuation and calibration of the Hes-ton and VG models

• using the initial calibration parameters throughout the risk horizon period. That is, not recalibrating the parameters and thus assuming the calibrated parameters give an ideal representation of the dynamics throughout the risk horizon period.

The first factor, which is termed calibration error, can be seen in Figures4.2and

4.4. The calibrated parameters for 30 and 90 days-to-maturity does not seem to

retrieve the market implied volatility smile for the Heston and VG models. The model risk regarding the use of semi-analytical formulae of the delta is implied in

Figure5.2.

Recall that the aim of the dissertation is to quantify the effect of the exponen-tial measure on VaR of the various portfolios and not to quantify the model risk

pertaining to each risk factor. Figures6.4 - 6.6 implicitly shows the effect of the

exponential change of measure. By analysing the difference between the nominal and the perturbed MC VaR for BS, Heston and VG, the biggest difference is found at 30 days-to-maturity. This could be due to the above-mentioned calibration error or the compounded effect from a combination of various risk factors.

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6.2 VaR model risk 32 0 50 100 150 200 250 300 Days to Maturity 0 1 2 3 4 5 6 95% VaR 104 =-1.5 =-0.5 =0.5 =2 Nominal 0 50 100 150 200 250 300 Days to Maturity -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Difference 104 =-1.5 =-0.5 =0.5 =2 (a)BS: 95% VaR 0 50 100 150 200 250 300 Days to Maturity 0 1 2 3 4 5 6 7 95% VaR 104 =-1.5 =-0.5 =0.5 =2 Nominal 0 50 100 150 200 250 300 Days to Maturity -1 0 1 2 3 4 5 Difference 104 =-1.5 =-0.5 =0.5 =2 (b)Heston 95% VaR 0 50 100 150 200 250 300 Days to Maturity 0 1 2 3 4 5 6 7 95% VaR 104 =-1.5 =-0.5 =0.5 =2 Nominal 0 50 100 150 200 250 300 Days to Maturity -1 0 1 2 3 4 5 Difference 104 =-1.5 =-0.5 =0.5 =2 (c)VG: 95% VaR

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6.2 VaR model risk 33 0 50 100 150 200 250 300 Days to Maturity -2.5 -2 -1.5 -1 -0.5 0 0.5 1 95% VaR 104 =-1.5 =-0.5 =0.5 =2 Nominal 0 50 100 150 200 250 300 Days to Maturity -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 Difference 104 =-1.5 =-0.5 =0.5 =2 (a)BS: 95% VaR 0 50 100 150 200 250 300 Days to Maturity -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 95% VaR 104 =-1.5 =-0.5 =0.5 =2 Nominal 0 50 100 150 200 250 300 Days to Maturity -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 Difference 104 =-1.5 =-0.5 =0.5 =2 (b)Heston 95% VaR 0 50 100 150 200 250 300 Days to Maturity -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 95% VaR 104 =-1.5 =-0.5 =0.5 =2 Nominal 0 50 100 150 200 250 300 Days to Maturity -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 Difference 104 =-1.5 =-0.5 =0.5 =2 (c)VG: 95% VaR

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Chapter 7 Discussion and Conclusion

In this section, the results throughout the dissertation are discussed and concluding remarks are presented.

The calibration results of the Black-Scholes, Heston and Variance-Gamma

mod-els were presented in Chapter4. Calibration of Black-Scholes, Heston and VG

re-flected European option prices similar to the market implied prices for options with times-to-maturity of [30, 90, 180, 270] days.

We implemented static delta hedging in order to obtain the model-dependent P&L distributions. The delta of the Heston and VG models was obtained from the semi-analytical formulae, whilst the closed-form analytical formula of the delta was used for the BS model. The resultant P&L distributions were presented in Section

6.1for each portfolio on BS, Heston and VG European options using four arbitrary

values of δ in order to analyse the effect of the exponential change of measure on the P&L distributions. The results suggest that δ can be interpreted as a pseudo risk-aversion measure. The effect of δ was shifting the P&L distributions, in particular

δ < 0shifted the distribution of the left and δ > 0 had the opposite effect. This was

seen across the various portfolios and days-to-maturity.

The effect of the exponential change of measure was observed on VaR of the

various portfolios and the results were presented in Section6.2. The exponential

change of measure had various effects on the worst-case VaR for each δ. The pseudo risk-aversion measure of δ = [0.5, 2] for ATM and long-only option portfolios across the BS, Heston and VG models illustrated the least deviation from the nominal model. In contrast, a pseudo risk-aversion measure of δ = [−0.5, −1.5] for the collar portfolio across the BS, Heston and VG models had the least deviation from the nominal model. This suggests that for regular option portfolios, one is afforded the flexibility of being less risk-averse without added financial implications.

Some risk factors contributing to the model error were identified using the

model risk classification framework (seeSchl ¨ogl(2016)). These include the use of

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Chapter 7. Discussion and Conclusion 35

sample paths without recalibrating to find the parameters describing each day. Re-call that model risk provides a framework that enables one to compare models and inherent risk. However the aim of this dissertation was not to quantify the model risk attributed to each risk factor. That is, the idiosyncratic effect for each type of model risk described by the model risk framework was not quantified.

In conclusion, quantification of the worst-case model error presents an alter-native method to model risk. This is evident in the results of VaR on the various portfolios. The use of δ allows one to quantify the pseudo risk-aversion measure and the precise value of the pseudo risk-aversion measure δ should be at the dis-cretion of the practitioner. Model risk quantification and the model risk framework are recommended as they illustrate the risks implicitly inherent in a model in com-parison to another model and presents a framework which classifies various risks. The model risk framework presents an opportunity for further investigation on alternative risk metrics such as Expected Shortfall or VaR on complex financial products, such as basket options. Complex option portfolios might be of regulatory interest as market data for these are not readily available. Regulatory bodies might be concerned with various internal models used to calculate historical VaR. These internal models might be subject to model error and quantifying the worst-case model error might be beneficial in developing a robust regulatory framework.

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BIBLIOGRAPHY 37

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Appendix A

Preliminaries

A.1 Definitions

Definition A.1. A metric, d(·, ·) : Rn_{× R}n_{→ R}+_{is defined as}

d(x, y) = kx − yk₂ =p(x1− y1)2+ · · · + (xn− yn)2, which satisfies the following conditions:

i d(x, y) ≥ 0, for all x, y ∈ Rn_,

ii d(x, y) = 0, if and only if x = y,

iii d(x, y) = d(y, x), for all x, y ∈ Rn_,

iv d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ Rn.

A.2 Lagrangian Duality

Consider the primal problem,

p0 = inf m f (m) subject to g(m) ≤ 0. (A.1) Define J (m) = ( f (m), if g(m) ≤ 0 ∞, else.

Then equationA.1is equivalent to p0= infmJ (m). Define the Lagrangian,

L(m, λ) := f (m) + λg(m). Then, sup λ≥0 L(m, λ) = ( f (m), if g(m) ≤ 0 ∞, else. = J (m),

and henceA.1is equivalent to p0 = infmsupλ≥0L(m, λ). The dual problem is

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A.2 Lagrangian Duality 39

have L(m, λ) ≤ J(m) for all λ, so that infmL(m, λ) ≤ J (m) = p0and hence d0 ≤ p0.

If the optimisation problem is convex, then d0 = p0.

Following similar reasoning as above, the primal problem is redefined as

p0 = inf

m f (m) subject to g(m), (A.2)

with the dual problem,

d0 = sup

λ∈R inf

m L(m, λ).

Similarly, a mixture primal of the form,

p0 = inf

m f (m) subject to g(m) ≤ 0 and h(m) = 0,

has the dual problem,

d0 = sup

λ≥0,µ∈R inf

m L(m, λ, µ),

where L(m, λ, µ) := f (m)+λg(m)+µh(m). Maximisation problems maxmf (m)

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Appendix B

B.1 Profit and Loss Distributions

-5 -4 -3 -2 -1 0 1 P&L 104 0 1 2 3 4 5 6 10 -4 _{BS: = 30days} Nominal = -1.5 = -0.5 = 0.5 = 2 -5 -4 -3 -2 -1 0 1 P&L 104 0 0.2 0.4 0.6 0.8 1 1.2 10 -3 _{Heston: = 30days} Nominal = -1.5 = -0.5 = 0.5 = 2 -6 -5 -4 -3 -2 -1 0 1 P&L 104 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 -4 _{VG: = 30days} Nominal = -1.5 = -0.5 = 0.5 = 2

Figure B.1:ATM portfolio nominal and perturbed P&L Distributions for the BS,

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B.1 Profit and Loss Distributions 41 -10 -8 -6 -4 -2 0 2 P&L 104 0 0.5 1 1.5 2 2.5 10 -4 _{BS: = 30days} Nominal = -1.5 = -0.5 = 0.5 = 2 -10 -8 -6 -4 -2 0 2 P&L 104 0 0.5 1 1.5 2 2.5 10 -4 _{Heston: = 30days} Nominal = -1.5 = -0.5 = 0.5 = 2 -10 -8 -6 -4 -2 0 2 P&L 104 0 0.5 1 1.5 2 2.5 10 -4 _{VG: = 30days} Nominal = -1.5 = -0.5 = 0.5 = 2

Figure B.2:Long-only portfolio nominal and perturbed P&L Distributions for the

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B.1 Profit and Loss Distributions 42 -1 0 1 2 3 4 5 6 P&L 104 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 10 -4 _{BS: = 30days} Nominal = -1.5 = -0.5 = 0.5 = 2 -1 0 1 2 3 4 5 6 7 P&L 104 0 0.5 1 1.5 2 2.5 3 3.5 4 10 -4 _{Heston: = 30days} Nominal = -1.5 = -0.5 = 0.5 = 2 -1 0 1 2 3 4 5 6 7 P&L 104 0 0.5 1 1.5 2 2.5 3 3.5 10 -4 _{VG: = 30days} Nominal = -1.5 = -0.5 = 0.5 = 2

Figure B.3:Collar portfolio nominal and perturbed P&L Distributions for the BS,

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B.1 Profit and Loss Distributions 43 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 P&L 104 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 10 -3 _{BS: = 180days} Nominal = -1.5 = -0.5 = 0.5 = 2 -5 -4 -3 -2 -1 0 1 P&L 104 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 10 -3 _{Heston: = 180days} Nominal = -1.5 = -0.5 = 0.5 = 2 -2.5 -2 -1.5 -1 -0.5 0 0.5 P&L 104 0 1 2 3 4 5 6 7 8 10 -3 _{VG: = 180days} Nominal = -1.5 = -0.5 = 0.5 = 2

Figure B.4:ATM portfolio perturbed P&L Distributions for the Black-Scholes (BS),

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B.1 Profit and Loss Distributions 44 -6 -5 -4 -3 -2 -1 0 1 P&L 104 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 -4 _{BS: = 180days} Nominal = -1.5 = -0.5 = 0.5 = 2 -5 -4 -3 -2 -1 0 1 P&L 104 0 1 2 3 4 5 6 10 -4 _{Heston: = 180days} Nominal = -1.5 = -0.5 = 0.5 = 2 -5 -4 -3 -2 -1 0 1 P&L 104 0 1 2 3 4 5 6 10 -4 _{VG: = 180days} Nominal = -1.5 = -0.5 = 0.5 = 2

Figure B.5:Long-only portfolio perturbed P&L Distributions for the Black-Scholes

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B.1 Profit and Loss Distributions 45 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 P&L 104 0 1 2 3 4 5 6 7 10 -4 _{BS: = 180days} Nominal = -1.5 = -0.5 = 0.5 = 2 -0.5 0 0.5 1 1.5 2 2.5 3 P&L 104 0 1 2 3 4 5 6 7 8 9 10 -4 _{Heston: = 180days} Nominal = -1.5 = -0.5 = 0.5 = 2 -1 0 1 2 3 4 5 6 P&L 104 0 0.5 1 1.5 2 2.5 10 -4 _{VG: = 180days} Nominal = -1.5 = -0.5 = 0.5 = 2

Figure B.6:Collar portfolio perturbed P&L Distributions for the Black-Scholes (BS),

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B.1 Profit and Loss Distributions 46 -2.5 -2 -1.5 -1 -0.5 0 0.5 P&L 104 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 10 -3 _{BS: = 270days} Nominal = -1.5 = -0.5 = 0.5 = 2 -7 -6 -5 -4 -3 -2 -1 0 1 P&L 104 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 10 -3 _{Heston: = 270days} Nominal = -1.5 = -0.5 = 0.5 = 2 -20000 -15000 -10000 -5000 0 5000 P&L 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 VG: = 270days Nominal = -1.5 = -0.5 = 0.5 = 2

Figure B.7:ATM portfolio perturbed P&L Distributions for the Black-Scholes (BS),

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B.1 Profit and Loss Distributions 47 -5 -4 -3 -2 -1 0 1 P&L 104 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 -4 _{BS: = 270days} Nominal = -1.5 = -0.5 = 0.5 = 2 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 P&L 104 0 1 2 3 4 5 6 7 10 -4 _{Heston: = 270days} Nominal = -1.5 = -0.5 = 0.5 = 2 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 P&L 104 0 1 2 3 4 5 6 7 10 -4 _{VG: = 270days} Nominal = -1.5 = -0.5 = 0.5 = 2

Figure B.8:Long-only portfolio perturbed P&L Distributions for the Black-Scholes

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B.2 VaR Results 48 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 P&L 104 0 1 2 3 4 5 6 7 8 10 -4 _{BS: = 270days} Nominal = -1.5 = -0.5 = 0.5 = 2 -0.5 0 0.5 1 1.5 2 2.5 P&L 104 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10 -3 _{Heston: = 270days} Nominal = -1.5 = -0.5 = 0.5 = 2 -2 -1 0 1 2 3 4 5 6 P&L 104 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 10 -4 _{VG: = 270days} Nominal = -1.5 = -0.5 = 0.5 = 2

Figure B.9:Collar portfolio perturbed P&L Distributions for the Black-Scholes (BS),

Heston and VG models 270 days-to-maturity options.

B.2 VaR Results

τ = 300 BS Heston VG

ATM Historical 5644.24 6287.62 6723.06

Monte Carlo 6157.33 189.17 7175.31

Long Only Historical 13045.07 13657.44 14058.28

Monte Carlo 14697.12 266.33 15764.18

Collar Historical 2853.12 3425.23 4061.27

Monte Carlo 2854.37 3435.95 4060.94

Quantifying Model Risk in Option Pricing and Value-at-Risk Models